Infinite Volume $\implies$ Infinite Cross-section

Question

Can one prove that if one has some geometric object in 3D Euclidean space (i.e. $\mathbb{R}^3$) and it has infinite volume, then it must have at least one cross-section that has infinite area?

Answer 1

This is not true.

Consider, for example, the union of all unit balls with centers on the curve $(t,t^2,t^3)$. This curve's perpendicular distance from an arbitary plane $ax+bx+cx=d$ is a non-constant polynomial in $t$, so the intersection between the tube and the plane is bounded (and thus finite).